Properties

Label 2-675-1.1-c3-0-73
Degree $2$
Conductor $675$
Sign $-1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.45·2-s + 11.8·4-s − 5.08·7-s + 17.3·8-s − 58.3·11-s − 21.2·13-s − 22.6·14-s − 17.8·16-s − 68.8·17-s − 40.8·19-s − 259.·22-s − 144.·23-s − 94.5·26-s − 60.3·28-s + 220.·29-s + 291.·31-s − 218.·32-s − 307.·34-s − 260.·37-s − 182.·38-s + 169.·41-s + 438.·43-s − 692.·44-s − 643.·46-s − 255.·47-s − 317.·49-s − 252.·52-s + ⋯
L(s)  = 1  + 1.57·2-s + 1.48·4-s − 0.274·7-s + 0.765·8-s − 1.59·11-s − 0.452·13-s − 0.432·14-s − 0.278·16-s − 0.982·17-s − 0.492·19-s − 2.51·22-s − 1.30·23-s − 0.713·26-s − 0.407·28-s + 1.40·29-s + 1.68·31-s − 1.20·32-s − 1.54·34-s − 1.15·37-s − 0.776·38-s + 0.646·41-s + 1.55·43-s − 2.37·44-s − 2.06·46-s − 0.792·47-s − 0.924·49-s − 0.672·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 4.45T + 8T^{2} \)
7 \( 1 + 5.08T + 343T^{2} \)
11 \( 1 + 58.3T + 1.33e3T^{2} \)
13 \( 1 + 21.2T + 2.19e3T^{2} \)
17 \( 1 + 68.8T + 4.91e3T^{2} \)
19 \( 1 + 40.8T + 6.85e3T^{2} \)
23 \( 1 + 144.T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 - 291.T + 2.97e4T^{2} \)
37 \( 1 + 260.T + 5.06e4T^{2} \)
41 \( 1 - 169.T + 6.89e4T^{2} \)
43 \( 1 - 438.T + 7.95e4T^{2} \)
47 \( 1 + 255.T + 1.03e5T^{2} \)
53 \( 1 - 214.T + 1.48e5T^{2} \)
59 \( 1 + 331.T + 2.05e5T^{2} \)
61 \( 1 - 54.9T + 2.26e5T^{2} \)
67 \( 1 + 758.T + 3.00e5T^{2} \)
71 \( 1 - 904.T + 3.57e5T^{2} \)
73 \( 1 + 866.T + 3.89e5T^{2} \)
79 \( 1 - 206.T + 4.93e5T^{2} \)
83 \( 1 + 463.T + 5.71e5T^{2} \)
89 \( 1 + 601.T + 7.04e5T^{2} \)
97 \( 1 + 229.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00445114373535151214387643167, −8.653895123171495304630688470262, −7.71156658833001926476947611776, −6.62203805076771838156904969524, −5.92056388232809856038357131665, −4.88989995733891607714094941092, −4.29309479827068826488386546601, −2.95478437740811664520517183444, −2.28720332740287806150101776029, 0, 2.28720332740287806150101776029, 2.95478437740811664520517183444, 4.29309479827068826488386546601, 4.88989995733891607714094941092, 5.92056388232809856038357131665, 6.62203805076771838156904969524, 7.71156658833001926476947611776, 8.653895123171495304630688470262, 10.00445114373535151214387643167

Graph of the $Z$-function along the critical line